Integralism symbol

DIALOGUE INTEGRALISM ON INTEGRA LISM SYMBOL

BY ARMAHEDI MAHZAR http://integralisme.wordpress.com

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Dialogue on integralism symbol Part One Ni Suiti and Ki Algo is my Anima and Animus which are the feminine and the masculine unconscious subpersonalities within myself. They are always in continuous conversation. even if I am in the front of one of the million eyes of the giant Tenretni and fingering her YTREWQ fingers.

In the following, is the dialogue of them concerning the picture that I put as myself in the my blog integralist.multiply.com. The Q(uestioner) is Ni Suiti and the A(nswerer) is Ki Algo. Hopefully you can all enjoy it.

Here we go!.

Q: What is the picture that Arma used as his blog identification?

A: That’s a part of a geometric pattern in the cover of his first book : Integralism.

Q: What geometrical pattern?

A: Aperiodic tesselation.

PERIODIC TESSELATION: CRYSTALS Q: What is tesselation anyway?

A: Tesselation is the covering of infinite plane with a finite set of tiles. If the tesselation periodic, from infinitely many possible regular polygons

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only three can tile a plane periodically: triangle, square and regular hexagon. Periodic tesselation means that you can shift the tiling pattern translationally to get the same pattern We have 3-fold rotationally symmetric tesselation

and the 4-fold rotationally symmetric tesselation

and the six-fold rotationally symmetric tesselation

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There are exactly 17 possible tilings of the plane with 3, 4 and 6-fold rotation symmetry. 13 of them can be found in Alhambra (I see it in here ) The combination of triangles, squares and hexagons can make a periodic tesselation such as

Q: Beautiful!. All of them is periodic, meaning you can shift the pattern and get the same pattern.

I see it in many mural decoration in islamic mosques and palaces and it can also be found naturally in the structure of all crystals. But your pattern is not periodic.

APERIODIC TESSELATION: FROM 20.426 to 2 A: That’s why I call it aperiodic, following the tradition of mathematics literature.

Q: Why do you interest in such tile?

A: Well, an article in Scientific American in the 60s was caught in Arma’s eyes. It was written by the logician Hao Wang. He asked if an infinite set of finite kinds of 2-way square color domino can fully cover all the plane periodically. It is called tiling problem.

Q: But the tile in Arma’s integralist symbol are not square, right?

A: Please do not interrupt me. The story to follow is an amazing story of mathematical simplification.

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It began in 1966 when Robert Berger demonstrated that the question Wang asked is unanswerable.

Mathematically the periodic tiling problem is in fact not decidable. He proved that if a finite kinds of tile can be used to cover all the plane, then it can not be filled in periodically. Such tiling called aperiodic. To prove it, he construct very large set of prototiles consisting 20,426 prototiles. He showed that that they can tiles a plane aperiodically. Fortunately, he was able to reduce the number of aperiodic tile to relatively small set containing 104 tiles.

Following Berger discovery, there is a rush of simplifications of the prototiles set. For example in 1968 Donald Knuth was

able to reduce the number to 92, then Robinson reduced it in 1971 to 35.

Roger Penrose improved it to 34. Robinson made more improvement by reducing to 24. Karel Culik II finally in 1996 gave an example of the set of 13 tiles

which tile the plane aperiodically.

By allowing rotation, Robinson had been able to reduced the number of prototile to just 6 tiles as it is shown in the following

Amazingly, three years later in 1974, the well known UK physicist, Roger Penrose had been able to reduce further the total number of required prototiles to just 2.

The trick is to change the shape of the tiles from squares to rhombi. Well for me it is very impressive, reduction from more than 20,000 to only 2 in less than a decade.

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The two rhombi are the thick and the thin ones.

Let us call them Thicky and Thinny

By joining the edges of thickies and thinnies we can form the following tesselation covering the infinite plane

Q: See, it is interesting but it is not the same as your integralism symbol.

A: Well, the Penrose rhombi can be each cut in half and rejoined edge by edge to form these Penrose Kite and Dart tiles. Joining two halves of Thinny makes the Dart and joining two halves of Thicky makes the Kite:

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we can form the following aperiodic tesselation.

I hope you will see that my integralist symbol is nothing but a subconfiguration of the above tesselation.

Q: OK. I see it now.

It is wonderful. It is Mathematical.

Mathematics is beautiful.

But it is really unnatural.

There is no such pattern in the crystal.

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Well, I could hardly hear what Ki Algo said in answering Ni Suiti. I think he was mumbling, and there was long silence after that.

I will try harder later to eavesdrop their conservation and report it to you to be enjoyed. See you next time.

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Dialogue on integralism symbol Part Two Ni Suiti is my Anima who loves visual arts, chemistry and geometry and Ki Algo is my Animus who loves music, physics and algebra.

They are still discussing my integralism symbol in the cover of my first book which is factually a Penrose tiling. In the following dialogue S is for Suiti and A is for Algo. Let us see, what are the facts known by Ki Algo on the realization of Penrose tiles in nature.

PHYSICAL REALIZATION: QUASICRYSTALS S: Arma’s symbol for integralism is a Penrose tiling and it is beautiful alright, but it is just a mathematical game. I think it will not appear in nature such as the 17 patterns of periodic tiles appear in crystals. It is forbidden. Am I wrong?

A: Well, you are wrong. It is true that there are only 17 patterns of crystal symmetry, none of them have 5-fold rotational symmetry, but exactly one year after the publication of Arma’s first book Integralism, the physicist Dan Schechtman announced the discovery of a phase of an Aluminium Manganese alloy which produced tenfold symmetric X-ray diffraction photograph. It may be similar to this photograph.

This symmetric pattern can only be explained if the atoms are arranged aperiodically in the form of three dimensional generalization of Penrose tiling as it is discovered by the amateur mathematician Robert Ammann.

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S: So, I am sure that they are not crystals. What are they?

A: Steinhardt and Levine, shortly after the announcement of Shechtman’s discovery, used the term ‘quasicrystal’ By the end of the 1980s the idea of quasicrystal became acceptable and in 1991 the International Union of Crystallography amended its definition of crystal, reducing it to the ability to produce a clear-cut diffraction pattern and acknowledging the possibility of the ordering to be either periodic or aperiodic.

With the new definition, quasicrystal is just a kinds of crystal: the aperiodic crystal.

S: How many types of quasicrystal are there?

A: In the same year of Schechtman publication, Ishimasa and coauthors published their discovery of twelvefold symmetry diffraction pattern of Ni-Cr particles.

Soon another equally challenging case presented a sample which gave a sharp eightfold diffraction picture. Lately, a team led by An Pang Tsai from Japan’s National Research Institute for Metals in Tsukuba has discovered quasicrystals of cadmium- ytterbium that are stable and exhibit three-dimensional icosahedral symmetry.

So there are four types of quasicrystals: •

8-fold or octagonal symmetric quasicrystals such as lV-Ni-Si Cr-Ni-Si Mn-Si Mn-Si-Al Mn-Fe-Si

•

10-fold or decagonal symmetric quasicrystals such as

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Al-Ni-Co Al-Cu-Mn Al-Cu-Fe Al-Cu-Ni Al-Cu-Co Al-Cu-Co-Si Al-Mn-Pd V-Ni-Si Cr-Ni

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12-fold or dodecagonal symmetric quasicrystals such as Cr-Ni V-Ni V-Ni-Si

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•

Dodecahedral symmetric quasicrystals such as Al-Mn Al-Mn-Si Al-Li-Cu Al-Pd-Mn Al-Cu-Fe Al-Mg-Zn Zn-Mg-RE Nb-Fe V-Ni-Si Pd-U-Si

S: Beautiful, but what’s the use of quasicrystals technologically?

A: Until now there are several applications, for example • • • •

thin film quasicrystal coating are used as a non-stick surface for saucepans. razor blades and surgical instruments are made from quasicrystalline material Quasicrystals have also been associated with hydrogen storage Metallic alloys are reinforced by deposition of icosahedral particles to improve the materials ability to withstand strain

S: I wonder if quasicrystal patterns also occur in cultural artifacts in history?

A: I think you can asked Arma’s friend Tenretni immediately I think we have to wait what Ni Suiti find as her answer.

Bye, for now.

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Dialogue on integralism symbol Part Three In the previous dialogue, Ki Algo was explaining mathematical and physical aspects of my integralism symbol. In the following dialogue, Ni Suiti report her findings of the cultural realizations of the aperiodic tiling in history. Listen!

CULTURAL REALIZATION: ISLAMIC ARCHITECTURE A: Hello Suiti. Do you have some answers from my giant fried Tenretni.

S: Yes, she told me that the physicist Peter Lu from Harvard University did some field research in Iran, Turkey, Azerbaijan and India and found a surprising fact that Islamic maths was 500 years ahead. See ABC News in Science webpage http://www.abc.net.au/science/news/stories/2007/1855313.htm?ancient .

A: That’s big news, but the data is too little to be significant.

S: There are so many discoveries To convince you I will list some of the strange ancient artifacts chronologically ordered. The decoration of the artifact is in the leftside and the aperiodic pattern is in the rightside. •

The Gunbad-i Kabud tomb tower in Maragha, Iran (1197

CE.),

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•

Abbasid Al-Mustansiriyya Madrasa in Baghdad, Iraq (1227-34 AD),

•

The Ilkhanid Uljaytu Mausoleum in Sultaniya, Iran (1304 AD),

•

The Mamluk Quran of Sandal (1306-15 AD)

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The Mamluk Quran of Aydughdi ibn Abdallah al-Badri (1313 AD),

•

The Timurid Tuman Aqa Mausoleum in the Shah-i Zinda complex in Samarkand, Uzbekistan (1405 AD).

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• •

the Ottoman Green Mosque in Bursa, Turkey (1424 AD),

•

The shrine of Khwaja Abdullah Ansari at Gazargah in Herat, Afghanistan (1425 to 1429 C.E.) (3, 9),

•

The Darb-i Imam shrine in Isfahan, Iran (1453 CE)

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•

The Friday Mosque, Isfahan, Iran (late 15th century AD).

A: Wow! So many forms in ancient artifacts structurally similar to the postmodern physics discoveries. Where do you find such data?

S: The above information are from supporting online material for the article in the magazine Science 315 (2007), 1106-1110 by Peter J. Lu and Paul J. Steinhardt, Decagonal and quasi-crystalline tilings in medieval Islamic architecture, The article is also available online in one of the author webpage http://www.physics.harvard.edu/~plu/publications/ A: I think, only scientists will read the article.

S: Oh no, the findings are so surprising, so it was reported all around the world in newspaper, magazines, radio and television broadcasting • • • •

Firstly in Lu’s campus http://www.news.harvard.edu/gazette/2007/03.01/99-tiles.html But there is also an article in New YorkTimes http://www.nytimes.com/2007/02/27/science/27math.html?hp Another in Newsweek International http://www.msnbc.msn.com/id/17553752/site/newsweek/ The BBC news in bahasa Indonesia is in http://www.bbc.co.uk/indonesian/news/story/2007/02/070223_geometricislamicart.shtml

The list for other worldwide news on the discovery can be found in Peter Lu webpage http://www.physics.harvard.edu/~plu/research/islamic_quasicrystal/ 16

A: Oh my goodness. I have never expected such explosion of news.

It’s very exciting. I’ve never known that art, mathematics and physics has such a common underlying form.

I hope you enjoy my report of eavesdropping Ki Algo and Ni Suiti dialogues.

I will report more on their dialogues, if I find something interesting to be shared in cyberspace.

See you later

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Dialogue on integralism symbol: Part FDour

In the last dialogue, Ni Suiti was explaining her discoveries in the answers of the shegiant Tenretni about the aperiodic pattern by the ancient muslim architects. It surprised Ki Algo. The following dialogue is their philosophical reflection ignited by such surprising discovery.

A: Why did aperiodic tiling has so many realizations? The concept, as mathematical entity, is realized in the mind of mathematicians. Afterward, it was discovered that it is realized physically in quasicrystals.

S: Astonishingly, it has also been realized in the mind of muslim architects for about five centuries. Because it is discovered in nature, then it should be there before it exists in the mind of mathematician or even the architects. How can it be?

A: Looking at these facts, I think the aperiodic tiling pattern is outside human mind and it is also outside natural world. I suppose that it must be in some kind of Plato’s World of Ideas of Mathematical Forms.

S: Well, if it is really there in the outside, it seems to me that it can be intelectually ‘seen’ with our mind’s ‘eyes’ mathematician sees them because they use their intuition. With the same intuition, they accept undefined concepts and underived axioms of any mathematical formal system such as geometry and arithmetics. The traditional muslim architects saw it there first because their remembrance practice, so their intuitive eyes are clear so the “remember” the form first before the mathematician. Plato called it ”anamnesis” A: Well, reflecting more deeply, I see that your analogy begging a question. If our mind can 18

‘see’ the form behind the thinks we see, can the mind also “see” the “Forms” behind the forms behind the things with our innermost eye?

S: The eyes that is behind the mind eyes behind the physical eyes arethe spiritual eyes. But what is the FORM behind the Mathematical “Forms” behind the physical forms?

A: Let me guess. All geometric forms are generated by using simple rules. I think there is the “Forms” behind the forms. I can discover it with logic. You can say that I “see” it with my mind’s “eyes.” But I can see it deeper. I see a FORM exists behind all the logical generating rules namely all your ‘Forms.’ It is structured set of relations between all the rules. It is the principle behind the rules. For such formal rules, the principle is symmetry. Periodicity is just one kind of symmetry. In fact, the aperiodicty of quasicrystals can be described as the projection of periodicity in higher dimensional space.

S: Periodicity in space is like musical rythm in time. In music, the rythm is the framework for the melody. The counterpart of melody in space is the mutual transformations of material forms. That’s it the mathematical forms is forming a the great symphony within what the ancient muslim philosopger called the knowledge of God. A: That’s a good metaphor. But what is the real reality?

S: The symmetry is the mathematical version of Beauty. The Beauty is the ultimate value beyond our world but penetrates to our world. Philosophers like to call such beyondness as transcendence and such penetrateness as immanence. Religious person call the Ultimate Transcendence as God. Beauty is just one of the characteristic of the Ultimate Transcendence.

A: What are the other ones?

S: The other characteristics of the ultimate transcendence is Truth and Goodness. Without Truth we can not get science. Without Goodness we can not have dynamically ordered society. The Goodness in society is Justice. If the Beauty manifest in the symmetry, The Truth manifests in the consistency and the Good manifests itself in the optimality.

A: What an enlightening vision!

S: Beauty, Truth and Goodness as the Atributes of the Ultimate Being are universal entities 19

which are realized differently in different cultures. But they are also realized universally and naturally in different forms of matter as natural Symmetry, Consistency and Optimality.

A: How can we see such Universal Trio?

S: The important thing is to know how do we “see� Beauty, Truth and Goodness. Symmetric Beauty can be realized by emotion or feeling, Consistent Truth by reason or Logic and Optimal Goodness by intuition.

A: Yes, I think we see such immaterial things through feeling, reason and intuition. But all those immaterial things is revealed in material things which we can see with sensation.

S: The late psychologist Carl Gustaf Jung said that those are the fundamental psychological function: (1) sensation, (2) feeling, (3) thinking and (4) intuition. Arma think that they are correlated to (1) matter, (2) energy, (3) information and (4) values respectively. We sense matter, feel energy, think information and intuit values. So, the fundamental psychological functions are correlated to the four categories of relative substances in integralism.

A: What Arma did not k now is that you are the archetype of Intuition and I am the archetype of Logic. He also did not know that your grand-daughter Si Nessa and my grand-son Si Emo is the archetypes the archetype of Sensation and Feeling respectively.

It is surprising to me that beside the two eldery archetypes, Ni Suiti and Ki Algo, there are two child archetypes, Si Nessa and Si Emo, who are accompanying them.

Pondering on the names of those archetypes, I finally found out that their names are the anagrams of indonesian terms related to the four psychological function that was listed

by Carl Gustaf Jung. Wow! Can you find it? See you later.

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Afternote on integralism symbol:

Dan Shechtman had obtained his Ph.D. from Technion – Israel Institute of Technology, and in 1983, he managed to get Ilan Blech, a colleague at his alma mater, interested in his findings of the "forbidden" 10-fold symmetric diffraction pattern.

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Together they attempted to interpret the diffraction pattern and translate it to the atomic pattern of a crystal. They submitted an article to the Journal of Applied Physics in the summer of 1984.

But the article came back seemingly by return of post – the editor had refused it immediately.

Shechtman then asked John Cahn, a renowned physicist who had lured him over to NIST in the first place, to take a look at his data. The otherwise busy researcher eventually did, and in turn, Cahn consulted with a French crystallographer, Denis Gratias.

In November 1984, together with Cahn, Blech and Gratias, Shechtman finally got to publish his data in Physical Review Letters. The article went off like a bomb among crystallographers. It questioned the most fundamental truth of their science: that all crystals consist of repeating, periodic patterns.

After a long waiting time, the Nobel Prize in Chemistry 2011 was finally awarded to Dan Shechtman "for the discovery of quasicrystals".

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